math 224 — Complex Arithmetic 1

1. The complex numbers

Definition The complex number system C is the set of all items of the form x + iy where x, y are real numbers, and i is a special symbol whose meaning will be described below.

It is useful to note that each nonzero real number x has both a magnitude |x| and a direction (the sign of x). There are only two directions on the number line, so there are only two choices of direction for a real number. We will see that a nonzero complex number z can be viewed as being a magnitude z and a direction in the plane. Here there are infinitely many different directions (one for each angle between 0 and 2π). Just as we have to worry about both magnitude and direction when multiplying two real numbers, we will worry about magnitudes and directions when multiplying two complex numbers. In fact multiplication by the complex number i (i.e., 0+1i) turns out to have no effect on magnitude, and changes the direction by ‘making a right angle turn counterclockwise’. The effect of 2 consecutive such turns is to reverse direction, so it is not surprising that i2 = −1. Note that the set of complex numbers can be identified with the plane R2 by the identification x + yi → (x, y). The difference between C and R2 is that there is extra algebraic structure on C, as we will describe below. Note that the complex number x + iy can be graphed in the plane as the point (x, y) Here is some notation: • We often use a single symbol (like ‘z’) to denote a complex number, and we write things like ‘z = x + yi’ to denote this.

• Any real number x can be thought of as the complex number x + 0i; we will also refer to the complex zero (0 + 0i) merely as ‘0’.

• If z is x + yi then the real part of z is Re(z) ≡ x, and the imaginary part of z is Im(z) ≡ y. Note that the real and the imaginary parts of z are both real numbers; the imaginary part does not include the i! If a complex number x + iy is graphed as (x, y) in the plane, the horizontal axis measures the real part, and the vertical axis measures the imaginary part.

• Two complex numbers are equal if and only if they have the same real parts and the same imaginary parts.

• We add two complex numbers by adding their real parts to get the real part of the sum and adding their imaginary parts to get the imaginary part of the sum:

if z = x + yi and w = s + ti then z + w = (x + s) + (y + t)i.

Similarly for subtraction.

• We multiply two complex numbers by multiplying everything out and making the identifica- tion i2 = −1. math 224 — Complex Arithmetic 2

We give the formula for multiplication below, but there is little point in trying to remember it; instead remember ‘multiply out, replace i2 by −1, and collect terms’.

if z = x + yi and w = s + ti then zw = (x + yi)(s + ti) = xs + xti + ysi + yti2 = (xs − yt) + (xt + ys)i.

• Before we can discuss division of complex numbers (which is slightly more complicated than multiplication) it is useful to introduce the notion of the complex conjugate of a complex number. The complex number that is obtained from z by changing the sign of its imaginary part is called the complex conjugate of z, and is denoted as z = x − iy.

• Some elementary, yet simple facts about complex conjugates are summarized in the following proposition.

Proposition 1.1 Let z = x + iy and w = u + iv be any two complex numbers, where x, y, u, v are real numbers. (a) z + w = z + w. z + z (b) Re(z) = . 2 z − z (c) Im(z) = . 2i (d) zz = x2 + y2 is a real number, and is strictly positive unless z = 0. (e) For any two complex numbers z, w, (zw) = (z)(w).

The proofs of these 5 assertions are left as exercises. (One way of establishing (e) is just to write z = x + iy, w = u + iv, compute both sides of the equation and see that they are equal. Alternatively, note that changing the signs of the imaginary parts of both z and w does not affect the real part of their product, and the only effect on the imaginary part of their product is that the sign changes.) • The fact that the product of a complex number z and its conjugate is a nonzero real number (unless z = 0) gives us a fairly simple way of dividing a nonzero complex number into another. The idea is simple; if z = x + iy is a complex number and r is a real number, then dividing z by r should be the same as multiplying z by 1/r, which would give (for z = z +yi) z/r = (x/r) + (y/r)i.

Now if w = u + iv is a complex number, then in order to compute z/w we can first replace w by a real number by multiplying both the numerator and the denominator by the conjugate of w z zw zw = = w ww u2 + v2 math 224 — Complex Arithmetic 3

where the denominator is now a real number, so we can proceed as above. For example 2 + 13i (2 + 13i)(3 + 4i) (6 − 52) + i(39 + 8) 48 47 = = = − + i . 3 − 4i (3 − 4i)(3 + 4i) 9 + 16 25 25 √ • Finally, the length of z = x + yi is defined to be |z| = x2 + y2. Note that |z| ≥ 0 with |z| = 0 if and only if z = 0, that |z| = |z|, and that |z|2 = zz.

1.1 Polar notation Just as we sometimes find it useful to use polar coordinates in the plane R2, it is sometimes useful to use polar notation in C. In polar notation we keep track of a complex number z by keeping track of its length |z| and the angle, measured counterclockwise, from the positive real axis to the segment from 0 through z. This angle is called the argument of z and is denoted by arg(z). Note that there is some ambiguity in the argument, because, depending on how they went about measuring this angle, one person might come up with a certain value θ while another comes up with θ + 2π. (As an example, one might claim that the argument of z = −i is −π/2 while another claims that it is 3π/2.) Rather than trying to stamp out this ambiguity we choose to live with it, and keep in mind that arg(z) is only specified up to some integer multiple of 2π. Now suppose that z ≠ 0. Then we can write |z| z z = z · = · |z| = |z|u |z| |z| where u is a complex number of length 1 (u = z/|z|), so that u lies on the unit circle in the plane. Clearly u and z have the same argument, which we will call θ, so that u = cos(θ) + i sin(θ). In short we can write z in polar notation as

z = |z|[cos(θ) + i sin(θ)] where arg(z) = θ.

An important observation about polar notation is the following: In multiplying two complex numbers, you multiply their lengths and you add their arguments. Here is the verification, where θ = arg(z) and α = arg(w)

zw = (|z|[cos(θ) + i sin(θ)] · |w|[cos(α) + i sin(α)]

= |z||w|[cos(θ) cos(α) − sin(θ) sin(α) + i(cos(θ) sin(α) + sin(θ) cos(α))] = |z||w|[cos(θ + α) + i sin(θ + α)] where we have used the addition formulas for sine and cosine to get the last line. In short

zw = |z||w|[cos(θ + α) + i sin(θ + α)] (1)

One important consequence of the last formula is that it shows that in taking powers of a complex number z we are taking powers of its length and multiples of its argument: the length of math 224 — Complex Arithmetic 4 zn is |z|n and the argument of zn is nθ where θ is the argument of z. It follows that in computing an nth root we will be taking roots of lengths and dividing arguments by n. The fact that arguments are multiplied by taking powers and divided by taking roots suggests that, somehow, the argument of a complex number is ‘living in an exponent’ (because√ exponents multiply under powers — (ab)n = anb — and exponents divide under roots — n ab = ab/n). In the next subsection we will try to justify the formula eiθ = cos(θ) + i sin(θ) which exactly describes the way in which the argument ‘lives in an exponent’, and is the key to understanding a number of important aspects of complex arithmetic. Using this formula, we see that z = |z|ei arg(z) is a true formula for any complex number z. The right side in the equation above is called the polar form of z, or the polar notation for z. Multiplication, division, powers, and roots are all straightforward for complex numbers written in polar form:

(reiα)(Reiβ) = rRei(α+β) r (reiα)/(Reiβ) = ei(α−β) R (reiα)n = rneinα √ √ n reiα = n reiα/n.

1.2 The complex exponential and Euler’s formula Consider the IVP z′ = ız, z(0) = 1, where z(t) is a function taking values in the complex plane, so z(t) = x(t) + ıy(t). Thus ız = −y + ıx. In the more familiar R2 notation, the IVP is x′ = −y, y′ = x, with (x(0), y(0)) = (1, 0). Equivalently, [ ] [ ][ ] [ ] [ ] x′ 0 −1 x x(0) 1 = , with = . y′ 1 0 y y(0) 0 It is easy to guess, and to check, that the solution is [ ] [ ] x cos(t) = , y sin(t) or in complex notation, z(t) = cos(t) + ı sin(t). Recall that if c is a real number, then the solution to x′ = cx with x(0) = 1 is x = ect. By analogy, it seems reasonable to define the solution of z′ = ız, z(0) = 0 to be the complex exponential function z = eıt. Since we have already seen that this solution is cos(t) + ı sin(t). We have arrived at the basic version of what is called Euler’s formula, namely

eıt = cos(t) + ı sin(t). math 224 — Complex Arithmetic 5

In the same way, if we define eıbt to be the solution of the IVP z′ = ıbz with z(0) = 1, then we arrive at a slightly more extended version of Euler’s formula,

eıbt = cos(bt) + ı sin(bt).

Finally, if we use this and calculate

[eateıbt]′ = aeateıbt + ıbeateıbt = (a + ıb)eateıbt,

we see that z = eateıbt solves the IVP z′ = (a + ıb)z with z(0) = 1, and so it makes sense to define

ea+ıb)t = eateıbt = eat[cos(bt) + ı sin(bt)],

which is the most general version of Euler’s formula. Actually, all that one needs to remember to reconstruct this last version is the most basic version, which is eıc = cos(c) + ı sin(c), along with the fact that the familiar rule for exponentials, ep+q = epeq, also works when p and q are complex numbers. 1.3 de Moivre’s theorem The relationship between the complex exponential and the sine and cosine give a very simple proof of a powerful result which is known as De Moivre’s theorem. Theorem 1.2

[cos(θ) + i sin(θ)]n = cos(nθ) + i sin(nθ).

Proof: The left side is (eiθ)n = einθ which is equal to the right side. 2

Note that the theorem gives a simple way of computing multiple-angle formulas: for instance, the cosine of 5θ is just the real part of [cos(θ) + i sin(θ)]5 so that (skipping a little arithmetic)

cos(5x) = cos5(x) − 10 cos3(x) + 5 cos(x).

1.4 Exercises

1. For z = 3 − 4i and w = 2 + 7i, write each of the following in the form a + ib.

(a) z + w. (b) 2w − 3z. (c) w. (d) |w|. (e) iz. math 224 — Complex Arithmetic 6

(f) zw. (g) w3. (h) w/z. (i) z/w. (j) Re(z), Im(z), Re(w), Im(w). 2. This problem is intended to help you visualize the geometric effect in the complex plane of some of the basic operations of complex arithmetic. (a) For z, w as in the last problem, graph each of the following points: z, w , z, w, z + w, 2z, iz. (b) For each row in the table below, describe (in a complete sentence or two) how the location of the point listed in the first column is related to the location of the point or points in the second column. z + w z, w z z 2z z iz z

3. For z and w as in exercise 1, find arg(z) and arg(w), and write each of z, w in the form reiθ where r > 0 and θ are real numbers. Use your answer to write w3 in this same form, then use the formula eiθ = cos(θ) + i sin(θ) to verify that your answer agrees with the answer to question 1(g). 4. Using your results from the last question, write zw in polar form. 5. Establish the truth of the 5 assertions of Proposition 1.1. 6. Verify the formula for cos(5x) that was given just after Theorem 1.2, and find a similar formula for sin(6x). 7. Formally calculate with the Taylor series for sin(t), cos(t), and eit so see that eit = cos(t) + ∑∞ − n 2n+1 ı sin(t). (The power series for sin(x), cos(x), and ex are sin(x) = ( 1) x , cos(x) = ∑ ∑ n=0 (2n+1)! ∞ (−1)nx2n x ∞ xn n=0 (2n)! , and e = n=0 n! .) 8. Show that (p, q) ∈ R2 has polar coordinates r and θ, then the point with polar coordinates r and −θ is (p, −q). Conclude that complex conjugates have opposite arguments. 9. Follow the following steps to show that any real-valued function of the form x(t) = p cos(kt) + q sin(kt) can be rewritten as x(t) = M cos(kt − α), (2) where the polar coordinates of the point (p, q) ∈ R2 are r = M, θ = α. math 224 — Complex Arithmetic 7

(a) Show that if w is the complex number p − ıq, then in polar notation w = Me−ıα. (See problem 8.) (b) Show that Re[(p − ıq)eıkt] = p cos(kt) + q sin(kt). (Use Euler’s Formula to expand eıkt.) (c) Show that Re[(p − ıq)eıkt] = M cos(kt − α). (Hint: use (a).) (d) Conclude that p cos(kt) + q sin(kt) = M cos(kt − α).

Thus x(t) is a sinusoidal oscillation of amplitude M, period 2π/k, and with a phase shift represented by α.

10. Show that if z = x + ıy is any complex number, then ız is another complex number of the same length as z, and also that as vectors in the plane, these two vectors are perpendicular.

11. Show that ız is just z rotated counterclockwise by π/2. (This is the geometric interpretation of the equation ı2 = −1: ‘make two consecutive left turns and you have turned around’).